We study the quantum-mechanical corrections to two point particles accelerated by a strut in a 2+1 D flat background. Since the particles are accelerating, we use finite temperature techniques to compute the Green's function of a conformally coupled scalar applying transparent and Dirichlet boundary conditions at the location of the strut. We find that the induced energy-momentum tensor diverges at the position of the strut unless we impose transparent boundary conditions. Further, we use the regular form of the induced energy-momentum tensor to calculate the gravitational backreaction on the original space. The resulting metric is a constant $\phi$ section of the 4-dimensional C-metric, and it describes two black holes corrected by weakly coupled CFT and accelerating in asymptotically flat spacetime. Interestingly enough, the same form of the metric was obtained before in 0803.2242 by cutting the AdS C-metric with angular dependent critical 2-brane. According to AdS/CFT+gravity conjecture, the latter should describe strongly coupled CFT black holes accelerating on the brane. The presence of the CFT at finite temperature gives us a unique opportunity to study the AdS/CFT+gravity conjecture at finite temperatures. We calculate various thermodynamic parameters to shed light on the nature of the strongly coupled CFT. This is the first use of the duality in a system containing accelerating particles on the brane.